On changes of measure in stochastic volatility models

1Mathematical Sciences Institute, The Australian National University, Canberra 0200, ACT, Australia2School of Actuarial Studies, University of New South Wales, Sydney 2152, NSW, Australia3Department of Statistics, Columbia University, New York 10027, NY, USA

Abstract

Pricing in mathematical finance often involves taking expected values under
different equivalent measures. Fundamentally, one needs to first ensure the
existence of ELMM, which in turn requires that the stochastic exponential of
the market price of risk process be a true martingale. In general, however,
this condition can be hard to validate, especially in stochastic volatility
models. This had led many researchers to “assume the condition away,” even though the condition is not innocuous, and nonsensical results can occur if it is in fact not satisfied. We provide an applicable theorem to check the conditions for a general class of Markovian stochastic volatility models. As an example we will also provide a detailed analysis of the Stein and Stein and Heston stochastic volatility models.

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